Borel laws on a separable Hilbert space are determined by basis coordinates

If $H=\{0\}$, then it has only one Borel probability measure and the conclusion holds. Otherwise, let $\mathcal{C}$ be the family of all finite coordinate cylinder sets ${\pi }_F^{-1}(B)$, where $F\subseteq J$ is finite and $B\subseteq {\mathbb{R}}^F$ is Borel. Each cylinder is Borel by continuity in [F1]. The family contains $H$, and the intersection of two cylinders is the finite-coordinate cylinder obtained on the union of their index sets. Hence $\mathcal{C}$ is a pi-system.

Enumerate the basis if it is infinite, and in the finite case repeat the last partial sum after all basis vectors have appeared. For $x_0\in H$ and $r>0$, put $S_N(x)={\sum }_{j\le N}\mathrm{\mid }⟨x-x_0,e_j⟩{\mathrm{\mid }}^2$. By [F2], $\mathrm{\parallel }x-x_0{\mathrm{\parallel }}^2={\sup }_N{S}_N(x)$, so $B(x_0,r)={\bigcup }_{\begin{array}{c}q\in \mathbb{Q}\\ 0\le q<r^2\end{array}}{\bigcap }_{N\ge 1}\{x:S_N(x)\le q\}\mathrm{.}$ Indeed, if $\mathrm{\parallel }x-x_0{\mathrm{\parallel }}^2<r^2$, choose a rational strictly between these two numbers; conversely, membership in one intersection gives $\mathrm{\parallel }x-x_0{\mathrm{\parallel }}^2\le q<r^2$. By [F1] and [F6], each $S_N$ is Borel, and $\{S_N\le q\}$ is a cylinder over a Borel subset of ${\mathbb{R}}^F$. Thus every open ball belongs to $\sigma (\mathcal{C})$.

Let $D$ be the set of all finite rational linear combinations of basis vectors. If the basis is finite, $D$ is at most countable by [F3]. If it is countably infinite, enumerate it and write $D$ as the union over $N$ of the sets of rational linear combinations of the first $N$ vectors. Each of those sets is at most countable by [F3], so their union is at most countable by the Countable Choice clause of [F3]. The real span of the basis is dense by completeness. Each finite real linear combination is approximated by a rational one: approximate each of its finitely many coefficients by a rational and use the triangle inequality and $\mathrm{\parallel }{e}_j\mathrm{\parallel }=1$. Thus $D$ is dense in $H$. By definition, this makes $H$ separable. Separability: the existence of an at most countable dense subset The balls $B(d,s)$ with $d\in D$ and positive rational $s$ form an at most countable family by [F3], and they are open by [F5]. They are a base: given $x\in U$ with $U$ open, choose $r>0$ with $B(x,r)\subseteq U$, then $d\in D$ with $\mathrm{\parallel }d-x\mathrm{\parallel }<r\mathrm{/}3$, and a rational $s$ with $\mathrm{\parallel }d-x\mathrm{\parallel }<s<r-\mathrm{\parallel }d-x\mathrm{\parallel }$. The triangle inequality gives $x\in B(d,s)\subseteq B(x,r)\subseteq U$. By step 2.1 every such ball lies in $\sigma (\mathcal{C})$, so every open set, as a union of a subfamily of this countable base, lies in $\sigma (\mathcal{C})$.

Every coordinate map is continuous by [F1], so $\sigma (\mathcal{C})\subseteq \mathcal{B}(H)$. Step 3.1 and the definition of the Borel sigma-algebra give the reverse inclusion. Thus $\sigma (\mathcal{C})=\mathcal{B}(H)$.

The assumed equality of finite-coordinate measures makes $\nu$ and ${\nu }^{\mathrm{\prime }}$ agree on every cylinder in $\mathcal{C}$. They are probability measures, so they agree on $H$ as well. The pi-system $\mathcal{C}$ generates $\mathcal{B}(H)$ by step 4.1; [F4] therefore gives $\nu ={\nu }^{\mathrm{\prime }}$.